EPSG Dataset :: v12.049

Mercator (variant A)

Coordinate Operation Method Details [VALID]

Name:

Mercator (variant A)

Code:

9804

Operation is Reversible:

Yes

Formula:

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The formulas to derive projected Easting and Northing coordinates are:

E = FE + a*ko(lon - lonO)
N = FN + a*ko* ln{tan(pi/4 + lat/2)[(1 - esin(lat))/(1 + esin(lat))]^e/2} where symbols are as listed above and logarithms are natural.

The reverse formulas to derive latitude and longitude from E and N values are:

lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 + 81e^8/1120) sin(6chi) + (4279e^8/161280) sin(8chi)

where chi = pi/2 - 2 arctan t
t = B^((FN-N)/(a*ko))
B = base of the natural logarithm, 2.7182818...
and for the 2 SP Case, ko is calculated as for the forward transformation above.
lon = ((E - FE)/(a*ko)) + lonO

Note: For the Mercator variant A (1SP) method, in the EPSG Dataset, to be fully transparent about the location of the projection origin, the parameter "latitude of natural origin (LatO)" is included in the defining parameters of the map projection method and map projections. It must have a value of zero because by definition the location of the natural origin for this method is on the equator. However this parameter is not used in the conversion formulas.

Example:

For Projected Coordinate System Makassar / NEIEZ

Parameters:
Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281
then e = 0.08169683

Latitude of natural origin = 00°00'00"N = 0.0000000 rad
Longitude of natural origin = 110°00'00"E = 1.91986218 rad
Scale factor at natural origin ko = 0.997
False Eastings FE = 3900000.00 m
False Northings FN = 900000.00 m

Forward calculation for:
Latitude = 3°00'00.00"S = -0.05235988 rad
Longitude = 120°00'00.00"E = 2.09439510 rad
gives
Easting E = 5009726.58 m
Northing N = 569150.82 m

Reverse calculation for same easting and northing first gives :
t = 1.0534121
chi = -0.0520110

Latitude = 3°00'00.000"S
Longitude = 120°00'00.000"E

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of natural origin	8801	No	The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
Longitude of natural origin	8802	No	The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the longitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0). Sometimes known as "central meridian (CM)".
Scale factor at natural origin	8805	No	The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the natural origin.
False easting	8806	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
False northing	8807	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.

Meta Data

Remarks:

Note that in these formulas the parameter latitude of natural origin (latO) is not used. However for this Mercator (variant A) method the EPSG dataset includes this parameter, which must have a value of zero, for completeness in CRS labelling.

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

November 4, 2021

Change ID:

[2001.08] [2004.43] [2008] [2010.058] [2021.033]

Alias:

Alias	Naming System	Remarks
Mercator (1SP)	EPSG alias	This was the method name used prior to October 2010.

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.